Energy decay estimates for the damped plate equation with a local degenerated dissipation

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Systems & Control Letters 48 (2003) 191 – 197 www.elsevier.com/locate/sysconle

Energy decay estimates for the damped plate equation with a local degenerated dissipation Rogelio Benavides Guzm,ana;∗;1 , Marius Tucsnakb a Universidade b Universit e

Federal do Rio de Janeiro, 22453 Rio de Janeiro, Brazil de Nancy I, 54506 Vandoeuvre les Nancy Cedex, France

Received 1 September 2001; received in revised form 27 March 2002; accepted 20 April 2002 Dedicated to the memory of J.L. Lions

Abstract We consider the Euler–Bernoulli plate equation in a bounded open set  of R2 with a degenerated local damping term. This dissipation is e7ective in a subset ! of  and the damping coe8cient may vanish in some subset of dimension one of !. We show that the usual observability inequality for the undamped problem implies polynomial decay estimates for the damped problem. Our method can be applied for other PDE’s such as the wave equation or the Schr
1. Introduction and statement of the main result Let  be a bounded open subset of R2 with boundary  and let ! be an arbitrary non-empty subset of . Let a = a(x) ∈ L∞ () be such that a(x) ¿ 0 for almost all x ∈  and suppose that
dx ¡ ∞; (1) p ! a for some p ¿ 0. We consider the following initial and boundary value problem modelling the damped vibrations of a ∗

Corresponding author. E-mail addresses: [email protected] (Rogelio Benavides Guzm,an), [email protected] (Marius Tucsnak). URL: http://www.iecn.u-nancy.fr/∼tucsnak 1 The work of this author was supported by the project ALFA of the European Community.

simply supported plate: y + 2 y + a(x)y = 0 y = Fy = 0

in  × (0; ∞);

on  × (0; ∞);

y(0) = y0

in ;

y (0) = y1

in :

(2)

The function a will be called “the localizing coe8cient” since it determines where the damping is e7ective in . The di8culty of the problem we consider consists in the fact that we don’t assume that the localizing coefIcient satisIes a condition of the form a(x) ¿ a0 ¿ 0 in !, as in most of the previous literature (cf. [16] and references therein). Localizing coe8cients satisfying only (1) were considered, in the case of the wave equation, in [12,14,11],

c 2002 Elsevier Science B.V. All rights reserved. 0167-6911/03/$ - see front matter  PII: S 0 1 6 7 - 6 9 1 1 ( 0 2 ) 0 0 2 6 4 - 5

192

R. Benavides Guzman, M. Tucsnak / Systems & Control Letters 48 (2003) 191 – 197

by applying multipliers techniques. As far as we know the present paper is the Irst one which tackles the plate equation with a local degenerated dissipation. Our method essentially relies on a classical observability inequality (see [9]) for the undamped problem and it can be applied for other partial di7erential equations (see [3] for applications to the wave and to the Schr
where




1 E(t) = 2



(|y (x; t)|2 + |Fy(x; t)|2 ) d x;

∀t ¿ 0;

(3)

is the energy of the plate. The above relation implies that for any T ¿ 0 we have
E(0) − E(T ) = a(x)|y (x; t)|2 d x dt: (4) ×(0;T )

In order to formulate our results we consider the Hilbert space H = [H 2 () ∩ H01 ()] × L2 (); s

where by H (); s ∈ R we denote the usual Sobolev spaces. We endowe this space with the norm  2  
 y  y   2 2 ∈ H:   = (|Fy| + |z| ) d x; ∀  z  z  H

We also consider the linear operator A : D(A) → H deIned by     v u A = ; v − 2 u − av

∀

  u v

∈ D(A);

(5)

where

 4       ∈ H () and D(A) =  = F = 0 on    ∈ H 1 () ∩ H 2 () 0

  

(6)

 

Moreover we need the domain of the powers of the operator A deIned recursively by D(Am ) = {U ∈ D(Am−1 )|AU ∈ D(Am−1 )}; m

∀m ¿ 2:

m

The operator A : D(A ) → H is as usually deIned by Am U =A(Am−1 U ), for all U ∈ D(Am ). It is well-known D(Am ) is a Hilbert space when equipped with the graph norm of Am . Some useful properties of the operator A deIned above are summarized in the result below. Since these properties are either well-known or very easy to prove (see, e.g., [4]), this result is stated without proof. Proposition 1. Let  ⊂ R2 be an open bounded set with smooth boundary (say C ∞ ) or let  be a rectangle. Moreover suppose that m ¿ 1. Then the following assertions hold true: (1) The operator A de>ned by (5) is the generator of a strongly continuous semigroup of contractions in H . P then we have the imbedding (2) If a ∈ C 2m−2 () D(Am ) ⊂ H 2m+2 () × H 2m ();

(7)

and the graph norm on D(Am ) is equivalent to the restriction to D(Am ) of the usual norm in H 2m+2 () × H 2m (). (3) The operator A generates a semigroup of linear contractions in D(Am ). As a consequence of the result above we obtain the following existence and regularity property. Corollary 2. Suppose that  and a satisfy the as0 sumptions in Proposition 1 and that ( yy1 ) ∈ D(Am ). Then the initial and boundary value problem (2) admits a unique solution y such that   m

y ∈ C m−j ([0; ∞); D(Aj )):  y j=0 Moreover, for all j ∈ {0; 1; : : : ; m} we have   0    y(t)   y      6 ; ∀t ¿ 0:      y (t)  j  y1  j D(A ) D(A )

R. Benavides Guzman, M. Tucsnak / Systems & Control Letters 48 (2003) 191 – 197

Let us now consider the undamped problem corresponding to (2), i.e., the initial and boundary value problem    + F2  = 0;    |@ = F|@ = 0; (8)    0  1 (0) =  ;  (0) =  : The main result of this paper is: Theorem 3. Let m ¿ 1, let  ⊂ R2 be an open bounded set with smooth boundary or let  be a rectangle. Suppose that ! is an open subset of  P satis>es and that the positive function a ∈ L∞ () condition (1). Moreover, suppose that there exist a constant C ¿ 0 and a time T ¿ 0 such that every solution of (8) satis>es the inequality
T

| |2 ¿ C (|1 |2 + |F0 |2 ) d x; 0

!

 ∀

0

 

∈ [H 2 () ∩ H01 ()] × L2 ():

(9)

Then there exists a constant Lm ¿ 0 (depending on a−1 Lp (!) and on a L∞ () ) such that the solutions of (2) satisfy the estimate:  0   y  Lm   E(t) 6 ; ∀t ¿ 0; (10)   (1 + t)2pm  y1  m D(A ) for all

 0 y y1

On the other hand, if  is a disk, the results in [1,5,8] imply that the solutions of the damped Schr




1

193

∈ D(Am ).

Remark 4. In the particular case of  being a rectangle Ja7ard showed in [6] that (9) holds for all open set ! ⊂ . Taking in consideration the fact that assumption (1) implies that a(x) ¿ 0 for x in an open subset !1 ⊂  it follows that, in this case, the result in Theorem 3 is not sharp, since the solutions of (2) decay exponentially in the energy space. Remark 5. Suppose that a ¿ a0 ¿ 0 in !1 , where !1 is an open subset of !. According to a result in [10] the solutions of (2) decay exponentially if and only if the same property holds for the system obtained by replacing the plate equation by the Schr
2. Some preliminaries In this section we Irst give several results, which will be useful in the proof of the main Theorem 3. We Irst recall a version of the Gagliardo–Nirenberg inequality which is a particular case of Lemma 1 in [12]. As usual, we denote by H m () the Sobolev space formed by the functions in L2 () which have distributional derivatives, up to the order m, in L2 (). Lemma 4. Suppose that m ¿ 1 and that  ⊂ R2 is an open set with smooth boundary or that  is a rectangle. Then there exists a constant C ¿ 0, depending only on , such that # L∞ () 6 C # H 2m () # L1− 2 () ; where

∀# ∈ H 2m ;

= 1=2m.

We also need the following technical lemma, which can be proved by using Lemma 3.3 in [7] (see also Ammari and Tucsnak [2]). Lemma 5. Let ("k )k∈N be a sequence of positive real numbers satisfying "k+1 6 "k − C"2+% k+1 ;

∀k ¿ 0;

194

R. Benavides Guzman, M. Tucsnak / Systems & Control Letters 48 (2003) 191 – 197

where C ¿ 0 and % ¿ − 1 are constants. Then there exists a positive constant M such that "k 6

M ; (1 + k)1=1+%

∀k ¿ 0:

(11)

We next recall several classical results in control theory which give, in particular, the interpretation of (9) in the terms of system theory. Throughout this section, U , H and Y are Hilbert spaces, A : D(A) → X is the generator of a strongly continuous semigroup T = (Tt )t¿0 on H , B ∈ L(U; H ) is an input operator and C ∈ L(H; Y ) is an observation operator. De#nition 6. Let A be the generator of a strongly continuous semigroup T on H and let C ∈ L(H; Y ). The pair (A; C) is exactly observable in time T ¿ 0 if there exists a constant m ¿ 0 such that
0

T

CTt x 2U dt ¿ m x 2H

∀x ∈ H:

The invariance of observability under output feedback is known for bounded input and observation operators (and beyond). More precisely (see, for instance, [15, Remark 6.5]) the following result holds. Proposition 7. Suppose that A is the generator of a strongly continuous semigroup T on H , C ∈ L(H; Y ), B ∈ L(U; H ) and that K ∈ L(Y; U ). Then the pair (A; C) is exactly observable in time T if and only if the pair (A+BKC; C) is exactly observable in time T . By applying Proposition 7 to our speciIc context we obtain the following result. Proposition 8. Suppose that every solution of (8) satis>es the inequality (9). Then there exists a constant m ¿ 0 such that m( (Fy)(0) 2L2 () + y (0) 2L2 () )
6

0

T


!

|y |2 +


0

T




a|y |2 :

(12)

for all y ∈ C(0; T ; H 2 ()) ∩ C 1 (0; T ; L2 ()) satisfying (2).

Proof. DeIne H = [H 2 () ∩ H01 ()] × L2 (); and consider the operator A : D(A) → H deIned by (5) and (6). Moreover, deIne the input space U = L2 () and the input operator B : U → H by   0 Bu = √ ∀u ∈ U: au The output space Y is deIned by Y = L2 () × L2 (!); whereas the observation operator C is given by   √      a C = ∀ ∈ H: |! Finally we deIne the feedback operator K ∈ L(Y; U ) by     y y K =y ∀ ∈ Y: z z A simple calculation shows that        0  BKC = ∀ ∈ H: a The above relation combined to (5) implies that,         ∀ ∈ D(A); (A + BKC) = −F2  The above relation and (9) imply the exact observability in time T for the couple (A + BKC; C). This implies, by Proposition 7, that the couple (A; C) is exactly observable in time T which is exactly the conclusion (12). 3. Proof of the main result Proof of Theorem 3. We Irst tackle the case m = 1. Let y be the solution of (2). By the H
×(0;T )


×

×(0;T )

a|y |2+2=p

p=p+1 ; (13)

R. Benavides Guzman, M. Tucsnak / Systems & Control Letters 48 (2003) 191 – 197

and that

195

Relation (17) implies, by using (4), that







 2

!×(0;T )

|y | 6

a

!×(0;T )

−p


×

1=p+1

s∈[0;T ]

 2+2=p

×(0;T )

y (s) 1=p+1 ) E(0) 6 C sup ( y (s) 1=p+1 H2 L2 p=p+1

|y |

;

×(E(0) − E(T ))p=p+1 :

(14)

(18)

If we denote

where p ¿ 0 is such that (1) holds true. Relation (13), (14) combined to Proposition 8 imply that there exists a constant C, depending on a−1 Lp (!) and on a L∞ () , such that

   y(t) 2   F(t) =    y (t) 

y0 2H 2 () + y1 2L2 ()

then from Corollary 2 there exists a constant C ¿ 0 such that


6C Since

 2+2=p

×(0;T )

 0 y y1

p=p+1

a|y |

:

(15)

∈ D(A), by Corollary 2, we have that

y ∈ C([0; ∞); H 2 ()): By applying the Gagliardo–Nirenberg inequality given in Lemma 4 (for m = 1) we get that 

y (s) L∞ () 6 C y



1=2 1=2  (s) H 2 () y (s) L2 () ;

∀s ∈ [0; T ];

(16)

where C ¿ 0 is a constant depending only on . We next use again the notation E(t) =

2 1 2 ( y(t) H 2 ()

+ y



and we follow the pattern of the proof in [12, p. 13]. We notice that (15) combined to Proposition 8 implies that there exists a constant C ¿ 0 such that, for all  0 y ∈ D(A) y1 we have

×(0;T )

a|y |2

p=p+1 :

and

y (s) 2L2 () 6 CE(0) ∀s ¿ 0: The two above inequalities combined to (18) imply that [E(0)](2p+1=2(p+1)) 6 C[F(0)]1=2(p+1) (E(0) − E(T ))p=p+1 :

(17)

(19)

Since the function t → E(t) is non-increasing, relation (19) implies, after a simple calculation, that there exists a constant C1 ¿ 0 such that [E(T )]1+1=2p : [F(0)]1=2p

(20)

We follow now the method used in Russell [13]. Estimate (20) remains valid in successive intervals [kT; (k + 1)T ]; k = 0; 1; : : : : Moreover, by Corollary 2, we have F(t) 6 F(0), for all t ¿ 0. It follows that there exists a constant C2 ¿ 0 such that E((k + 1)T ) 6 E(kT ) − C1

"k =

E(kT ) ; F(0)

[E((k + 1)T )]1+1=2p ; [F(0)]1=2p (21)

If we deIne

s∈[0;T ]

×

∀t ¿ 0;

∀k ¿ 0:

E(0) 6 C sup ( y (s) 1=p+1 y (s) 1=p+1 ) H2 L2 


y (s) 2H 2 () 6 CF(0);

E(T ) 6 E(0) − C1

(t) 2L2 () );

;

D(A)

k = 0; 1; : : :

196

R. Benavides Guzman, M. Tucsnak / Systems & Control Letters 48 (2003) 191 – 197

then (21) can be rewritten as ; "k+1 6 "k − C2 "1+1=2p k+1

∀k ¿ 0:

By applying Lemma 5 with % = (1=2p) − 1 we obtain the existence of a constant M ¿ 0 such that  0 2  y  M   E(kT ) 6 ; ∀k ¿ 0:   (1 + k)2p  y1  D(A)

Using again the fact that E is non-increasing, the above relation implies that there exists a constant L1 ¿ 0, depending on a−1 Lp (!) and on a L∞ () , such that  0 2  y  L1   ; ∀t ¿ 0: E(t) 6   2p (1 + t)  y1  D(A) We have thus obtained our main result in the case m = 1. For m ¿ 2 the result can be obtained by induction over m. More precisely, suppose that n ¿ 2 is such that the assertion in Theorem 3 holds for m ∈ {1; : : : ; n−1} and let  0 y ∈ D(An ): 1 y Then, by using the semigroup property, we have that    y(t) 2 Ln−1   ; E(2t) 6   (1 + t)2(n−1)p  y (t);  n−1 D(A ) ∀t ¿ 0:

(22)

for some constant Ln−1 ¿ 0. Moreover, there exists a constant C ¿ 0 such that     2    y(t) 2   n−1 y(t)   6 A   ;     y (t)  n−1 y (t) H D(A ) ∀t ¿ 0: If we denote     y(t) z(t) n−1 =A ; y (t) z  (t)

(23)

(24)

then z(t) is the solution of (2) with initial data  0 y n−1 ∈ D(A): A y1 Since we already proved the assertion of the theorem for m = 1 it follows that      z(t) 2  z(0) 2 L1     : (25)     6   z (t)  (1 + t)2p  z  (0)  H D(A) On the other hand from (24) it follows that   0    z(0)   y      6C  ;      z (0)   y1  n D(A) D(A )

(26)

for some constant C ¿ 0. Relations (23), (25) and (26) imply that the conclusion of the theorem also holds for m = n. We have thus ended our proof.

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